On Sun, 04 Dec 2022 14:22:40 -0500, Richard Damon wrote:

> On 12/4/22 10:20 AM, olcott wrote:
>> On 12/4/2022 3:31 AM, Mikko wrote:
>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>
>>>> If simulating halt decider H correctly simulates its input D until H
>>>> correctly determines that its simulated D would never stop running
>>>> unless aborted then H can abort its simulation of D and correctly
>>>> report that D specifies a non-halting sequence of configurations.
>>>>
>>>> Once the above is verified to be a tautology then it can be accepted
>>>> as an axiom for Simulating halt deciders.
>>>
>>> Once that is verified to be a tautology then it can (and should) be
>>> removed from the set of axioms for Simulating halt deciders (and other
>>> axiom sets).
>>> It is regarded as a bad style to use redundant axioms, and tautologies
>>> are theorems of logic, thus redundant as axioms of a theory.
>>>
>>> Mikko
>>>
>>>
>> All axioms are tautologies. The only way to correctly determine that a
>> brand new axiom has been discovered is to verify that it is a
>> tautology.
>>
>>
> No, a tautology is a statment that by its structure MUST be true for all
> cases.
>
> An axiom is a statement of accepted self-evident truth. They are not
> "Discovered", except in the creation of a new branch of a field, as they
> form the basis of that branch/field.
>
> For example, Euclid's 5th Axiom: Given a line, and a point not on that
> line, only one line can be drawn through that point that never touches
> that line.
>
> This statement is taken as a true statement in Euclidian Geometry.
>
> That statement is NOT a tautology in general, as in non-Euclidian
> Geometry it does not hold.

That isn't Euclid's 5th Axiom at all: you spout more crap than that
deviant, Olcott who you cannot resist engaging with. You are suspect and
unreliable.

/Flibble

On 12/4/22 2:38 PM, Mr Flibble wrote:
> On Sun, 04 Dec 2022 14:22:40 -0500, Richard Damon wrote:
>
>> On 12/4/22 10:20 AM, olcott wrote:
>>> On 12/4/2022 3:31 AM, Mikko wrote:
>>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>>
>>>>> If simulating halt decider H correctly simulates its input D until H
>>>>> correctly determines that its simulated D would never stop running
>>>>> unless aborted then H can abort its simulation of D and correctly
>>>>> report that D specifies a non-halting sequence of configurations.
>>>>>
>>>>> Once the above is verified to be a tautology then it can be accepted
>>>>> as an axiom for Simulating halt deciders.
>>>>
>>>> Once that is verified to be a tautology then it can (and should) be
>>>> removed from the set of axioms for Simulating halt deciders (and other
>>>> axiom sets).
>>>> It is regarded as a bad style to use redundant axioms, and tautologies
>>>> are theorems of logic, thus redundant as axioms of a theory.
>>>>
>>>> Mikko
>>>>
>>>>
>>> All axioms are tautologies. The only way to correctly determine that a
>>> brand new axiom has been discovered is to verify that it is a
>>> tautology.
>>>
>>>
>> No, a tautology is a statment that by its structure MUST be true for all
>> cases.
>>
>> An axiom is a statement of accepted self-evident truth. They are not
>> "Discovered", except in the creation of a new branch of a field, as they
>> form the basis of that branch/field.
>>
>> For example, Euclid's 5th Axiom: Given a line, and a point not on that
>> line, only one line can be drawn through that point that never touches
>> that line.
>>
>> This statement is taken as a true statement in Euclidian Geometry.
>>
>> That statement is NOT a tautology in general, as in non-Euclidian
>> Geometry it does not hold.
>
> That isn't Euclid's 5th Axiom at all: you spout more crap than that
> deviant, Olcott who you cannot resist engaging with. You are suspect and
> unreliable.
>
> /Flibble

There are many forms that the statement is presented in, and all are
called Euclid's 5th Axiom.

Euclid didn't call them axioms, but postulates, as the theory of logic
was still evolving, so that wasn't the term he used, but is what is used
in modern studies of the theory.

What do YOU think Eucid's 5th Axiom is? and does it actually express
something different?

On 12/4/2022 1:22 PM, Richard Damon wrote:
> On 12/4/22 10:20 AM, olcott wrote:
>> On 12/4/2022 3:31 AM, Mikko wrote:
>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>
>>>> If simulating halt decider H correctly simulates its input D until H
>>>> correctly determines that its simulated D would never stop running
>>>> unless aborted then H can abort its simulation of D and correctly
>>>> report that D specifies a non-halting sequence of configurations.
>>>>
>>>> Once the above is verified to be a tautology then it can be accepted
>>>> as an axiom for Simulating halt deciders.
>>>
>>> Once that is verified to be a tautology then it can (and should) be
>>> removed
>>> from the set of axioms for Simulating halt deciders (and other axiom
>>> sets).
>>> It is regarded as a bad style to use redundant axioms, and
>>> tautologies are
>>> theorems of logic, thus redundant as axioms of a theory.
>>>
>>> Mikko
>>>
>>
>> All axioms are tautologies. The only way to correctly determine that a
>> brand new axiom has been discovered is to verify that it is a tautology.
>>
>
> No, a tautology is a statment that by its structure MUST be true for all
> cases.
>

A tautology is any expression of language that is verified as completely
true entirely on the basis of its meaning.

Any expression of language that is a tautology can be used as if it was
an axiom in that in both cases it is known to be true thus forming a
sound basis for inference.

> An axiom is a statement of accepted self-evident truth. They are not
> "Discovered", except in the creation of a new branch of a field, as they
> form the basis of that branch/field.
>

Simulating halt deciders are the new field.
That the above paragraph is a tautology provides a stronger basis for
its truth than if it was merely assumed to be true as an axiom.

> For example, Euclid's 5th Axiom: Given a line, and a point not on that
> line, only one line can be drawn through that point that never touches
> that line.
>
> This statement is taken as a true statement in Euclidian Geometry.
>

So you are saying that it is false in other cases?

> That statement is NOT a tautology in general, as in non-Euclidian
> Geometry it does not hold.

--
Copyright 2022 Olcott

"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer

On 12/4/22 3:14 PM, olcott wrote:
> On 12/4/2022 1:22 PM, Richard Damon wrote:
>> On 12/4/22 10:20 AM, olcott wrote:
>>> On 12/4/2022 3:31 AM, Mikko wrote:
>>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>>
>>>>> If simulating halt decider H correctly simulates its input D until
>>>>> H correctly determines that its simulated D would never stop
>>>>> running unless aborted then H can abort its simulation of D and
>>>>> correctly report that D specifies a non-halting sequence of
>>>>> configurations.
>>>>>
>>>>> Once the above is verified to be a tautology then it can be
>>>>> accepted as an axiom for Simulating halt deciders.
>>>>
>>>> Once that is verified to be a tautology then it can (and should) be
>>>> removed
>>>> from the set of axioms for Simulating halt deciders (and other axiom
>>>> sets).
>>>> It is regarded as a bad style to use redundant axioms, and
>>>> tautologies are
>>>> theorems of logic, thus redundant as axioms of a theory.
>>>>
>>>> Mikko
>>>>
>>>
>>> All axioms are tautologies. The only way to correctly determine that a
>>> brand new axiom has been discovered is to verify that it is a tautology.
>>>
>>
>> No, a tautology is a statment that by its structure MUST be true for
>> all cases.
>>
>
> A tautology is any expression of language that is verified as completely
> true entirely on the basis of its meaning.
>

What your source of that definition?

From: https://en.wikipedia.org/wiki/Tautology_(logic)

In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula
or assertion that is true in every possible interpretation. An example
is "x=y or x≠y". Similarly, "either the ball is green, or the ball is
not green" is always true, regardless of the colour of the ball.

So,

> Any expression of language that is a tautology can be used as if it was
> an axiom in that in both cases it is known to be true thus forming a
> sound basis for inference.

Nope. Source for that statement?

>
>> An axiom is a statement of accepted self-evident truth. They are not
>> "Discovered", except in the creation of a new branch of a field, as
>> they form the basis of that branch/field.
>>
>
> Simulating halt deciders are the new field.

But must conform to all the requirmeent of Halt Deciders, or that aren't
Halt Decider.

> That the above paragraph is a tautology provides a stronger basis for
> its truth than if it was merely assumed to be true as an axiom.

But it isn't one (at least with your meaning of the words), so you whole
theory colapses.

>
>> For example, Euclid's 5th Axiom: Given a line, and a point not on that
>> line, only one line can be drawn through that point that never touches
>> that line.
>>
>> This statement is taken as a true statement in Euclidian Geometry.
>>
>
> So you are saying that it is false in other cases?

Yes.

You never heard of the Non-Euclidean Geometries?

For example, in Spherical Geometry, it is IMPOSSIBLE to place a
(straight) line through a point not on another line, that never touches
that other line.

>
>> That statement is NOT a tautology in general, as in non-Euclidian
>> Geometry it does not hold.
>

On 12/4/22 3:14 PM, olcott wrote:

> A tautology is any expression of language that is verified as completely
> true entirely on the basis of its meaning.
>

If so, then it is PROVABLE by the definitions of the words.

Of course to do this, you need to put yourself on record for the meaning
of the words, and need to be using them with the same defintion
everywhere, and a definition compatible with the things you are using in
your proof.

Since your definition of "A Correct Simulation" is going to fall flat
there, you have your problem.

To make the simulation actually show the behavior of the program, it
will need to include complete, but to show your H is "correct", it needs
to allow partial.

Thus, your "tautology" fails to be a true statement, so it CAN'T be a
Tautology.

YOU FAIL.

On 2022-12-04 12:22, Richard Damon wrote:
> On 12/4/22 10:20 AM, olcott wrote:
>> On 12/4/2022 3:31 AM, Mikko wrote:
>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>
>>>> If simulating halt decider H correctly simulates its input D until H
>>>> correctly determines that its simulated D would never stop running
>>>> unless aborted then H can abort its simulation of D and correctly
>>>> report that D specifies a non-halting sequence of configurations.
>>>>
>>>> Once the above is verified to be a tautology then it can be accepted
>>>> as an axiom for Simulating halt deciders.
>>>
>>> Once that is verified to be a tautology then it can (and should) be
>>> removed
>>> from the set of axioms for Simulating halt deciders (and other axiom
>>> sets).
>>> It is regarded as a bad style to use redundant axioms, and
>>> tautologies are
>>> theorems of logic, thus redundant as axioms of a theory.
>>>
>>> Mikko
>>>
>>
>> All axioms are tautologies. The only way to correctly determine that a
>> brand new axiom has been discovered is to verify that it is a tautology.
>>
>
> No, a tautology is a statment that by its structure MUST be true for all
> cases.
>
> An axiom is a statement of accepted self-evident truth. They are not
> "Discovered", except in the creation of a new branch of a field, as they
> form the basis of that branch/field.

I'd drop the 'self-evident' part. That might have been the view of the
Greeks, but I don't think people today would consider axioms to be
self-evident. Your example of the Fifth Postulate is a case in point:
while it might have seemed self-evident to Euclid, the existence of
non-Euclidian geometries suggest otherwise.

Axioms are simply statements which are taken as given in a particular
system.

André

> For example, Euclid's 5th Axiom: Given a line, and a point not on that
> line, only one line can be drawn through that point that never touches
> that line.
>
> This statement is taken as a true statement in Euclidian Geometry.
>
> That statement is NOT a tautology in general, as in non-Euclidian
> Geometry it does not hold.

--
To email remove 'invalid' & replace 'gm' with well known Google mail
service.

On Sunday, 4 December 2022 at 17:20:30 UTC+2, olcott wrote:
> All axioms are tautologies. The only way to correctly determine that a
> brand new axiom has been discovered is to verify that it is a tautology.
That's incoherent. A tautology is a statement which is "true in all models".

But the trivial case to manufacture a model in which the axiom is not true is to define the negation of the axiom as true.

x=x is not a tautology because not(x=x) can be defined as an axiom.

Trivial example: https://ibb.co/9yV0nBy

On Monday, 5 December 2022 at 04:18:36 UTC+2, richar...@gmail.com wrote:
> On 12/4/22 3:14 PM, olcott wrote:
> > A tautology is any expression of language that is verified as completely
> > true entirely on the basis of its meaning.
> >
> If so, then it is PROVABLE by the definitions of the words.
Uh. Prove that x=x "from the definition of the words".

> Of course to do this, you need to put yourself on record for the meaning
> of the words, and need to be using them with the same defintion
> everywhere, and a definition compatible with the things you are using in
> your proof.
To prove that something is a tautology requires you to prove it true in ALL models.

I have no idea how you are going to enumerate ALL models of any statement.

> Since your definition of "A Correct Simulation" is going to fall flat
> there, you have your problem.
You mean exactly like all of your definitions are going to fall flat?

That's not a problem with any particular definition. That's a problem with definability.

> To make the simulation actually show the behavior of the program, it
> will need to include complete, but to show your H is "correct", it needs
> to allow partial.
>
> Thus, your "tautology" fails to be a true statement, so it CAN'T be a
> Tautology.
>
> YOU FAIL.
But that's a problem with ALL Mathematics in general - not Olcott's decider in particular.

What's your point?

On Monday, 5 December 2022 at 02:58:44 UTC+2, richar...@gmail.com wrote:
> From: https://en.wikipedia.org/wiki/Tautology_(logic)
>
> In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula
> or assertion that is true in every possible interpretation. An example
> is "x=y or x≠y". Similarly, "either the ball is green, or the ball is
> not green" is always true, regardless of the colour of the ball.

Shame. Those are terrible examples of tautologies.

They are tautologies of classical logic (where excluded middle holds).
They are not tautologies of Intuitionistic logic.

So your "tautology" is not true in every model.

🤣🤣🤣🤣

On 12/5/22 2:14 AM, Skep Dick wrote:
> On Monday, 5 December 2022 at 02:58:44 UTC+2, richar...@gmail.com wrote:
>> From: https://en.wikipedia.org/wiki/Tautology_(logic)
>>
>> In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula
>> or assertion that is true in every possible interpretation. An example
>> is "x=y or x≠y". Similarly, "either the ball is green, or the ball is
>> not green" is always true, regardless of the colour of the ball.
>
> Shame. Those are terrible examples of tautologies.
>
> They are tautologies of classical logic (where excluded middle holds).
> They are not tautologies of Intuitionistic logic.
>
> So your "tautology" is not true in every model.
>
> 🤣🤣🤣🤣

Yes, I agree their examples make assumptions about the system they are
being expressed in.

After all, 'x = x', isn't a tautology in all systems.

The key point is Olcott seem to like to treat Wikipedia as a reliable
source, and they are giving a significantly different definition, it
shows that he needs to find a justification for the one he wants to use.

Especially since he draws an incorrect conclusion from his definition.

On 12/5/22 1:57 AM, Skep Dick wrote:
> On Sunday, 4 December 2022 at 17:20:30 UTC+2, olcott wrote:
>> All axioms are tautologies. The only way to correctly determine that a
>> brand new axiom has been discovered is to verify that it is a tautology.
> That's incoherent. A tautology is a statement which is "true in all models".
>
> But the trivial case to manufacture a model in which the axiom is not true is to define the negation of the axiom as true.
>
> x=x is not a tautology because not(x=x) can be defined as an axiom.
>
> Trivial example: https://ibb.co/9yV0nBy
>

Except that adding the axiom that negates the tautology doesn't
invalidate the logic that proved the tautology, it only makes the new
system with the added axiom inconsistent.

Remember, ALL statements are made in the context of a logic system that
implicitly imports all the definitions of that system. That system is
often just implicitly defined by the context of the conversation.

Yes, a Tautology will often only be a Tautology in the system it is
being talked about in.

On 12/5/22 2:05 AM, Skep Dick wrote:
> On Monday, 5 December 2022 at 04:18:36 UTC+2, richar...@gmail.com wrote:
>> On 12/4/22 3:14 PM, olcott wrote:
>>> A tautology is any expression of language that is verified as completely
>>> true entirely on the basis of its meaning.
>>>
>> If so, then it is PROVABLE by the definitions of the words.
> Uh. Prove that x=x "from the definition of the words".

Given that the statement must be coming from a system that is presumed
to be somewhat "normal", since there are system that define that some
members of their domain do not compare equal to themselvess even if they
are the same as them selves (Like the "Not-a-number" value of IEEE
floating point math.)

A typical prove for a normal system like this could go something like,
nornal system being a reasonable assumption since the convention is that
abnormal sustems need to be declaired before being used:

A variable like "x" when it appears multiple times in a given set of
expressions will refer to the same member of the domain for all occurrences.

Since the statement uses the "=" sign, which by default implies
equality, and the default rules of equality state that equality is
normally a reflexive operation (something is equal to itself)

Since the two "x"s by the first definintion refer to the same item, and
the "=" refers to equality, x = x is saying that some object is equal to
itself, which is a property of the presumed equality rule,

This statment is proven to be true in a normal domain.

>
>> Of course to do this, you need to put yourself on record for the meaning
>> of the words, and need to be using them with the same defintion
>> everywhere, and a definition compatible with the things you are using in
>> your proof.
> To prove that something is a tautology requires you to prove it true in ALL models.
>
> I have no idea how you are going to enumerate ALL models of any statement.

I guess that just shows your model of thinking can't handle infinities.

>
>> Since your definition of "A Correct Simulation" is going to fall flat
>> there, you have your problem.
> You mean exactly like all of your definitions are going to fall flat?
>
> That's not a problem with any particular definition. That's a problem with definability.

Nope. since you define things in a context, which ultimately gets you
back to those first principles that were just accepted as defined.

Yes, if you reject the concept of First Principles, you end up not being
able to know anything.
>
>> To make the simulation actually show the behavior of the program, it
>> will need to include complete, but to show your H is "correct", it needs
>> to allow partial.
>>
>> Thus, your "tautology" fails to be a true statement, so it CAN'T be a
>> Tautology.
>>
>> YOU FAIL.
> But that's a problem with ALL Mathematics in general - not Olcott's decider in particular.
>
> What's your point?
>

Can you prove THAT statement?

Or are you just showing the limitations of the systems you have stuck
yourself into.

On 2022-12-04 15:20:27 +0000, olcott said:

> On 12/4/2022 3:31 AM, Mikko wrote:
>> On 2022-12-03 17:34:55 +0000, olcott said:
>>
>>> If simulating halt decider H correctly simulates its input D until H
>>> correctly determines that its simulated D would never stop running
>>> unless aborted then H can abort its simulation of D and correctly
>>> report that D specifies a non-halting sequence of configurations.
>>>
>>> Once the above is verified to be a tautology then it can be accepted as
>>> an axiom for Simulating halt deciders.
>>
>> Once that is verified to be a tautology then it can (and should) be removed
>> from the set of axioms for Simulating halt deciders (and other axiom sets).
>> It is regarded as a bad style to use redundant axioms, and tautologies are
>> theorems of logic, thus redundant as axioms of a theory.
>>
>> Mikko
>>
>
> All axioms are tautologies. The only way to correctly determine that a
> brand new axiom has been discovered is to verify that it is a tautology.

Not necessarily, as Richard Damon already pointed out. Of course, if you
wish, you can call the rest "postulates" instead of axioms. What exactly
is the difference between an axiom and a postulate is a matter of opinion,
but the most common opinion is that there is no difference, or at least
no important difference, and the words "axiom" and "postulate" are
synonymes.

Mikko

On 2022-12-05 02:29:23 +0000, André G. Isaak said:

> On 2022-12-04 12:22, Richard Damon wrote:
>> On 12/4/22 10:20 AM, olcott wrote:
>>> On 12/4/2022 3:31 AM, Mikko wrote:
>>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>>
>>>>> If simulating halt decider H correctly simulates its input D until H
>>>>> correctly determines that its simulated D would never stop running
>>>>> unless aborted then H can abort its simulation of D and correctly
>>>>> report that D specifies a non-halting sequence of configurations.
>>>>>
>>>>> Once the above is verified to be a tautology then it can be accepted as
>>>>> an axiom for Simulating halt deciders.
>>>>
>>>> Once that is verified to be a tautology then it can (and should) be removed
>>>> from the set of axioms for Simulating halt deciders (and other axiom sets).
>>>> It is regarded as a bad style to use redundant axioms, and tautologies are
>>>> theorems of logic, thus redundant as axioms of a theory.
>>>>
>>>> Mikko
>>>>
>>>
>>> All axioms are tautologies. The only way to correctly determine that a
>>> brand new axiom has been discovered is to verify that it is a tautology.
>>>
>>
>> No, a tautology is a statment that by its structure MUST be true for all cases.
>>
>> An axiom is a statement of accepted self-evident truth. They are not
>> "Discovered", except in the creation of a new branch of a field, as
>> they form the basis of that branch/field.
>
> I'd drop the 'self-evident' part. That might have been the view of the
> Greeks, but I don't think people today would consider axioms to be
> self-evident. Your example of the Fifth Postulate is a case in point:
> while it might have seemed self-evident to Euclid, the existence of
> non-Euclidian geometries suggest otherwise.

Actually, Euclid did not consider his fifth postulate, as he called it,
self-evident. Other philosophers had proposed that it is possible, given
a straight line and a point outside it, draw from the point two asymptotic
straight lines so that they intersect each other but not the given line.
Euclid thought otherwise but failed to prove, so he had to make it
a postulate. In his constructions and proofs Eiclid avoided the use of
the fifth postulate as much as possible.

The difference between postulates and axioms, as Euclid used the words,
is that the postulates are specific to geometry but irrelevant to e.g.
accounting and axioms like "when eequals are added to equals the sums
are equal" are relevant to both and other purposes.

Mikko

On 12/5/2022 12:57 AM, Skep Dick wrote:
> On Sunday, 4 December 2022 at 17:20:30 UTC+2, olcott wrote:
>> All axioms are tautologies. The only way to correctly determine that a
>> brand new axiom has been discovered is to verify that it is a tautology.
> That's incoherent. A tautology is a statement which is "true in all models".
>

I have broadened the definition so that it equally applies to natural
language.

> But the trivial case to manufacture a model in which the axiom is not true is to define the negation of the axiom as true.
>

The axiom/tautology must be verified as true entirely on the basis of
its meaning.

> x=x is not a tautology because not(x=x) can be defined as an axiom.
>
> Trivial example: https://ibb.co/9yV0nBy
>

That is mere nonsense.
When we define that a herd of cows is a pile of bricks we are wrong.
When we ignore the law of non-contradiction we are wrong.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 12/5/2022 8:29 AM, Mikko wrote:
> On 2022-12-04 15:20:27 +0000, olcott said:
>
>> On 12/4/2022 3:31 AM, Mikko wrote:
>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>
>>>> If simulating halt decider H correctly simulates its input D until H
>>>> correctly determines that its simulated D would never stop running
>>>> unless aborted then H can abort its simulation of D and correctly
>>>> report that D specifies a non-halting sequence of configurations.
>>>>
>>>> Once the above is verified to be a tautology then it can be accepted
>>>> as an axiom for Simulating halt deciders.
>>>
>>> Once that is verified to be a tautology then it can (and should) be
>>> removed
>>> from the set of axioms for Simulating halt deciders (and other axiom
>>> sets).
>>> It is regarded as a bad style to use redundant axioms, and
>>> tautologies are
>>> theorems of logic, thus redundant as axioms of a theory.
>>>
>>> Mikko
>>>
>>
>> All axioms are tautologies. The only way to correctly determine that a
>> brand new axiom has been discovered is to verify that it is a tautology.
>
> Not necessarily, as Richard Damon already pointed out. Of course, if you
> wish, you can call the rest "postulates" instead of axioms. What exactly
> is the difference between an axiom and a postulate is a matter of opinion,
> but the most common opinion is that there is no difference, or at least
> no important difference, and the words "axiom" and "postulate" are
> synonymes.
>
> Mikko
>

With axioms and tautologies both are necessarily true thus impossibly
false. We can know that a herd of cows is not a pile of bricks on the
basis of the meaning of the words. These meanings are expressed as
stipulated relations between finite strings.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 12/5/2022 5:01 AM, Richard Damon wrote:
> On 12/5/22 2:14 AM, Skep Dick wrote:
>> On Monday, 5 December 2022 at 02:58:44 UTC+2, richar...@gmail.com wrote:
>>> From: https://en.wikipedia.org/wiki/Tautology_(logic)
>>>
>>> In mathematical logic, a tautology (from Greek: ταυτολογία) is a formula
>>> or assertion that is true in every possible interpretation. An example
>>> is "x=y or x≠y". Similarly, "either the ball is green, or the ball is
>>> not green" is always true, regardless of the colour of the ball.
>>
>> Shame. Those are terrible examples of tautologies.
>>
>> They are tautologies of classical logic (where excluded middle holds).
>> They are not tautologies of Intuitionistic logic.
>>
>> So your "tautology" is not true in every model.
>>
>> 🤣🤣🤣🤣
>
> Yes, I agree their examples make assumptions about the system they are
> being expressed in.
>
> After all, 'x = x', isn't a tautology in all systems.
>
> The key point is Olcott seem to like to treat Wikipedia as a reliable
> source, and they are giving a significantly different definition, it
> shows that he needs to find a justification for the one he wants to use.
>
> Especially since he draws an incorrect conclusion from his definition.

I am unifying (thus broadening) these concepts across formal systems and
natural language to derive the most basic foundation of the notion of
analytical truth.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On Monday, 5 December 2022 at 13:01:46 UTC+2, richar...@gmail.com wrote:
> After all, 'x = x', isn't a tautology in all systems.
If 'x=x' isn't a tautology then saying "in all systems" is redundant.

You know. By definition.

On 12/5/2022 7:47 AM, Mikko wrote:
> On 2022-12-05 02:29:23 +0000, André G. Isaak said:
>
>> On 2022-12-04 12:22, Richard Damon wrote:
>>> On 12/4/22 10:20 AM, olcott wrote:
>>>> On 12/4/2022 3:31 AM, Mikko wrote:
>>>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>>>
>>>>>> If simulating halt decider H correctly simulates its input D until
>>>>>> H correctly determines that its simulated D would never stop
>>>>>> running unless aborted then H can abort its simulation of D and
>>>>>> correctly report that D specifies a non-halting sequence of
>>>>>> configurations.
>>>>>>
>>>>>> Once the above is verified to be a tautology then it can be
>>>>>> accepted as an axiom for Simulating halt deciders.
>>>>>
>>>>> Once that is verified to be a tautology then it can (and should) be
>>>>> removed
>>>>> from the set of axioms for Simulating halt deciders (and other
>>>>> axiom sets).
>>>>> It is regarded as a bad style to use redundant axioms, and
>>>>> tautologies are
>>>>> theorems of logic, thus redundant as axioms of a theory.
>>>>>
>>>>> Mikko
>>>>>
>>>>
>>>> All axioms are tautologies. The only way to correctly determine that a
>>>> brand new axiom has been discovered is to verify that it is a
>>>> tautology.
>>>>
>>>
>>> No, a tautology is a statment that by its structure MUST be true for
>>> all cases.
>>>
>>> An axiom is a statement of accepted self-evident truth. They are not
>>> "Discovered", except in the creation of a new branch of a field, as
>>> they form the basis of that branch/field.
>>
>> I'd drop the 'self-evident' part. That might have been the view of the
>> Greeks, but I don't think people today would consider axioms to be
>> self-evident. Your example of the Fifth Postulate is a case in point:
>> while it might have seemed self-evident to Euclid, the existence of
>> non-Euclidian geometries suggest otherwise.
>
> Actually, Euclid did not consider his fifth postulate, as he called it,
> self-evident. Other philosophers had proposed that it is possible, given
> a straight line and a point outside it, draw from the point two asymptotic
> straight lines so that they intersect each other but not the given line.
> Euclid thought otherwise but failed to prove, so he had to make it
> a postulate. In his constructions and proofs Eiclid avoided the use of
> the fifth postulate as much as possible.
>
> The difference between postulates and axioms, as Euclid used the words,
> is that the postulates are specific to geometry but irrelevant to e.g.
> accounting and axioms like "when eequals are added to equals the sums
> are equal" are relevant to both and other purposes.
I think you have it right. When I took geometry about 65+ years ago, my
teacher state that there where two classes of assumptions that were used
in geometry: the first were properties and relations of "geometric
objects" such as there is exactly one line that contains two given
points; the second was basically the properties of numbers as yours
above "equals added to equals are equals". I hadn't thought about that
distinction for a while.
--
Jeff Barnett

On Monday, 5 December 2022 at 15:16:36 UTC+2, richar...@gmail.com wrote:
> Except that adding the axiom that negates the tautology doesn't
> invalidate the logic that proved the tautology, it only makes the new
> system with the added axiom inconsistent.
You are confused about the whole thing. Tautology means true in ALL possible interpretations.

If a statement is true when interpreted in system A; but false when interpreted in system B then it's not a tautology! Because there is an interpretation in which it's NOT true.

It has nothing to do with the consistency of the system as Bolyai, Lobachevsky, and Riemann all have shown that going contrary to Euclid's parallel postulate doesn't yield any contradictions.

Euclud assumed one line could be drawn through a point parallel to a given line
Bolyai asumed many lines could be drawn through a point parallel to a given line.
Riemann assumed that no line could be drawn through a point parallel to a given line.

No contradiction - just three different geometries.

> Remember, ALL statements are made in the context of a logic system that
> implicitly imports all the definitions of that system. That system is
> often just implicitly defined by the context of the conversation.
That's an oxymoron. Context is always implicit, not explicit.

On 12/5/22 10:25 AM, olcott wrote:
> On 12/5/2022 5:01 AM, Richard Damon wrote:
>> On 12/5/22 2:14 AM, Skep Dick wrote:
>>> On Monday, 5 December 2022 at 02:58:44 UTC+2, richar...@gmail.com wrote:
>>>> From: https://en.wikipedia.org/wiki/Tautology_(logic)
>>>>
>>>> In mathematical logic, a tautology (from Greek: ταυτολογία) is a
>>>> formula
>>>> or assertion that is true in every possible interpretation. An example
>>>> is "x=y or x≠y". Similarly, "either the ball is green, or the ball is
>>>> not green" is always true, regardless of the colour of the ball.
>>>
>>> Shame. Those are terrible examples of tautologies.
>>>
>>> They are tautologies of classical logic (where excluded middle holds).
>>> They are not tautologies of Intuitionistic logic.
>>>
>>> So your "tautology" is not true in every model.
>>>
>>> 🤣🤣🤣🤣
>>
>> Yes, I agree their examples make assumptions about the system they are
>> being expressed in.
>>
>> After all, 'x = x', isn't a tautology in all systems.
>>
>> The key point is Olcott seem to like to treat Wikipedia as a reliable
>> source, and they are giving a significantly different definition, it
>> shows that he needs to find a justification for the one he wants to use.
>>
>> Especially since he draws an incorrect conclusion from his definition.
>
> I am unifying (thus broadening) these concepts across formal systems and
> natural language to derive the most basic foundation of the notion of
> analytical truth.
>
So, you ADMIT that you aren't working in the ACTUAL COMPUTATION THEORY
of the Halting Problem, so you are working on that Problem.

Note, you can't change something by "broadning" it, and remain working
with it if you change how things work in the original field.

You H is just WRONG about P(P), and saying that is just because you have
"broadened" the terms is just a way to try to disguise LYING,

As an example of CORRECT broadening, expanding the number system from
Naturals to Interger to Rational to Real, etc. In each case the
operations on the smaller set, when done on the members of the smaller
set, give the same results in the bigger set as they do when done in the
smaller set.

This doesn't happen in your "Broadening" of the Halting Problem.

On 12/5/22 12:30 PM, Skep Dick wrote:
> On Monday, 5 December 2022 at 13:01:46 UTC+2, richar...@gmail.com wrote:
>> After all, 'x = x', isn't a tautology in all systems.
> If 'x=x' isn't a tautology then saying "in all systems" is redundant.
>
> You know. By definition.

Nope, because a tautology is a statement of logic, which inplies a field
of operation with which to take its context.

The statement when taken out of context, needs to be put into a context
to have meaning.

The author of that example wasn't thinking about contexts where equality
isn't defined to one self in all cases, and it is a fairly odd case.

When put into a context as imagined by the author, it is a tautolgy.
There exists contexts where it isn't. It just shows that we do need to
be away of the context we are talking about, and avoid pulling
statements out of the area where the context is understood.

YOU have shown that you don't understand that statements can't have
actual meaning without context, which is why you can't know anything.

On 12/5/22 2:17 PM, Skep Dick wrote:
> On Monday, 5 December 2022 at 15:16:36 UTC+2, richar...@gmail.com wrote:
>> Except that adding the axiom that negates the tautology doesn't
>> invalidate the logic that proved the tautology, it only makes the new
>> system with the added axiom inconsistent.
> You are confused about the whole thing. Tautology means true in ALL possible interpretations.

Then nothing is a tautology, as you can also totally reinterprete the
meaning of all the symbols used in the statement.

Even 1 = 1 can't be tautology as = could be defined as being compose of
exactly the same atoms in the same place, and the two 1's are written in
different physical locations and thus not equal.

>
> If a statement is true when interpreted in system A; but false when interpreted in system B then it's not a tautology! Because there is an interpretation in which it's NOT true.
>
> It has nothing to do with the consistency of the system as Bolyai, Lobachevsky, and Riemann all have shown that going contrary to Euclid's parallel postulate doesn't yield any contradictions.
>
> Euclud assumed one line could be drawn through a point parallel to a given line
> Bolyai asumed many lines could be drawn through a point parallel to a given line.
> Riemann assumed that no line could be drawn through a point parallel to a given line.
>
> No contradiction - just three different geometries.

Which is about axioms, and not tautologies, so you have gone off track.

>
>> Remember, ALL statements are made in the context of a logic system that
>> implicitly imports all the definitions of that system. That system is
>> often just implicitly defined by the context of the conversation.
> That's an oxymoron. Context is always implicit, not explicit.
>

On 12/5/22 10:19 AM, olcott wrote:
> On 12/5/2022 8:29 AM, Mikko wrote:
>> On 2022-12-04 15:20:27 +0000, olcott said:
>>
>>> On 12/4/2022 3:31 AM, Mikko wrote:
>>>> On 2022-12-03 17:34:55 +0000, olcott said:
>>>>
>>>>> If simulating halt decider H correctly simulates its input D until
>>>>> H correctly determines that its simulated D would never stop
>>>>> running unless aborted then H can abort its simulation of D and
>>>>> correctly report that D specifies a non-halting sequence of
>>>>> configurations.
>>>>>
>>>>> Once the above is verified to be a tautology then it can be
>>>>> accepted as an axiom for Simulating halt deciders.
>>>>
>>>> Once that is verified to be a tautology then it can (and should) be
>>>> removed
>>>> from the set of axioms for Simulating halt deciders (and other axiom
>>>> sets).
>>>> It is regarded as a bad style to use redundant axioms, and
>>>> tautologies are
>>>> theorems of logic, thus redundant as axioms of a theory.
>>>>
>>>> Mikko
>>>>
>>>
>>> All axioms are tautologies. The only way to correctly determine that a
>>> brand new axiom has been discovered is to verify that it is a tautology.
>>
>> Not necessarily, as Richard Damon already pointed out. Of course, if you
>> wish, you can call the rest "postulates" instead of axioms. What exactly
>> is the difference between an axiom and a postulate is a matter of
>> opinion,
>> but the most common opinion is that there is no difference, or at least
>> no important difference, and the words "axiom" and "postulate" are
>> synonymes.
>>
>> Mikko
>>
>
> With axioms and tautologies both are necessarily true thus impossibly
> false. We can know that a herd of cows is not a pile of bricks on the
> basis of the meaning of the words. These meanings are expressed as
> stipulated relations between finite strings.
>

No, axioms are assumed true, and generally "self-evident".

Tautologies are statements provably true by the nature of the statement
and not needing emperical knowledge. It is true in ALL embodyments of
the basic system.

"Jack is twelve" is NOT a tautology, even if Jack happens to be twelve.

"If Jack was eleven a year ago, he now is twelve" would be a tautology
under the assumption that we are dealing with system with a normal
passage of time.

On Tuesday, 6 December 2022 at 07:55:15 UTC+2, richar...@gmail.com wrote:
> No, axioms are assumed true, and generally "self-evident".
>
> Tautologies are statements provably true by the nature of the statement
> and not needing emperical knowledge. It is true in ALL embodyments of
> the basic system.
Distinction without a difference.

The "law" of excluded middle is formalised as. ~p ∨ p ⇔ ⊤
In English that spells out "~p ∨ p" is materially equivalent to a tautology.

> "Jack is twelve" is NOT a tautology, even if Jack happens to be twelve.
But '"Jack is twelve" is true' is a tautology.

> "If Jack was eleven a year ago, he now is twelve" would be a tautology
> under the assumption that we are dealing with system with a normal
> passage of time.
That's most definitely not a tautology. It's a conditional.

You know, because ⊤ means "unconditionally true".